|Method||Ultra-relativistic regime||Handling of discontinuities||Extension to several spatial dimensions||
Since their introduction in numerical RHD at the beginning of nineties, HRSC methods have demonstrated their ability to describe accurately (stable and without excessive smearing) relativistic flows of arbitrarily large Lorentz factors and strong discontinuities, reaching the same quality as in classical hydrodynamics. In addition (as it is the case for classical flows, too), HRSC methods show the best performance compared to any other method (e.g., artificial viscosity, FCT or SPH).
Despite of the latter fact, a lot of effort has been put into improving these non-HRSC methods. Using a consistent formulation of artificial viscosity has significantly enhanced the capability of finite difference schemes  as well as of relativistic SPH  to handle strong shocks without spurious post-shock oscillations. However, this comes at the price of a large numerical dissipation at shocks. Concerning relativistic SPH, recent investigations using a conservative formulation of the hydrodynamic equations [30, 164] have reached an unprecedented accuracy with respect to previous simulations, although some issues still remain. Besides the strong smearing of shocks, the description of contact discontinuities and of thin structures moving at ultra-relativistic speeds needs to be improved (see Section 6.2).
Concerning FCT techniques, those codes based on a conservative formulation of the RHD equations have been able to handle relativistic flows with discontinuities at all flow speeds, although the quality of the results is below that of HRSC methods in all cases .
The extension to multi-dimensions is simple for most relativistic codes. Finite difference techniques are easily extended using directional splitting. Note, however, that HRSC methods based on exact solutions of the Riemann problem [109, 187] first require the development of a multidimensional version of the relativistic Riemann solver. The adapting-grid, artificial viscosity, implicit code of Norman & Winkler  and the relativistic Glimm method of Wen et al.  are restricted to one dimensional flows. Note that Glimm's method produces the best results in all the tests analyzed in Section 6 .
The symmetric TVD scheme proposed by Davis  and extended to GRMHD (see below) by Koide et al.  combines several characteristics making it very attractive. It is written in conservation form and is TVD, i.e., it is converging to the physical solution. In addition, it is independent of spectral decompositions, which allows for a simple extension to RMHD. Quite similar statements can be made about the approach proposed by van Putten . In contrast to FCT schemes (which are also easily extended to general systems of equations), both Koide et al.'s and van Putten's methods are very stable when simulating mildly relativistic flows (maximum Lorentz factors ) with discontinuities. Their only drawback is an excessive smearing of the latter. A comparison of Davis' method with Riemann solver based methods would be desirable.
|Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
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